Everton have been chasing the central defender this summer.
Manchester City's Sergio Aguero in action with Sunderland's Lamine Kone
The Blues are keen to further add to their defence this summer following the arrival of Ashley Williams as a replacement for John Stones this week.
Kone seems to be the man they have targeted, with the Liverpool Echo reporting they were confident of getting a deal done by the end of the week.
Sunderland's Lamine Kone in action with Crystal Palace's Jordon Mutch
Their chances appeared to increase this week when Sky Sports reported he had released a statement about his desire to leave the Stadium of Light this summer.
However, Sunderland are very much playing hard ball amid the interest, with Moyes determined to find a replacement before letting him go.
Lamine Kone scores the second goal for Sunderland
And according to the Mirror they have also increased his asking price to £20 million to ensure they get a good price for their man should he eventually leave.
It is unknown whether Everton are willing to pay that price or will instead turn their attention elsewhere.
Much could rest on youngster Mason Holgate, with him impressing in the draw with Tottenham and hinting the Blues may not actually need further recruits.
Sunderland's Lamine Kone celebrates with team mates after scoring their first goal